Hydrodynamic and Thermal Characterization of Steady MHD Flow in Channels and Pipes Considering Viscous Dissipation and Joule Heating

This study presents a comparative sensitivity analysis of the Hartmann number (Ha) and Brinkman number (Br) on magnetohydrodynamic (MHD) flow in rectangular channels and circular pipes. Normalized sensitivity coefficients quantify the response of key metrics, including velocity, wall shear stress, temperature, and convective heat transfer, with validation against recent experimental and numerical studies. The system equations were solved through a coupled analytical–numerical method coded in Python 3.14; velocity field was solved analytically whereas temperature field was discretized using a finite differences scheme and solved numerically using the Thomas algorithm. The entire code was written by the authors. The results show that Ha predominantly governs hydrodynamics, inducing velocity suppression, flow flattening, and enhanced wall shear stress. Rectangular channels experience stronger Hartmann layer effects, while circular pipes exhibit smoother velocity profiles. Conversely, Br primarily controls thermal behavior, with higher values intensifying internal heat generation and elevating centerline temperature, potentially attenuating the average Nusselt number at high Br levels. Nonlinear Ha–Br interactions define distinct operational regimes, from heat transfer enhancement to thermal degradation. Optimal performance windows are identified: Ha ≈ 8–12 and Br ≈ 0.05–0.3 for channels, and Ha ≈ 10–15 and Br ≈ 0.1–0.4 for pipes, balancing thermal and hydraulic efficiency. Deviations from benchmark studies remain within ±5%, confirming predictive reliability. This work provides practical design guidance for advanced MHD thermal systems and establishes a foundation for future studies on temperature-dependent properties, three-dimensional effects, and complex flow regimes.